Rudin's problem on groups and a generalization of mean value theorem
It is well known that, for (differentiable) real functions, \[ \frac{f(x)-f(y)}{x-y}= f'(\frac{x+y}{2})\tag{*} \] characterizes the polynomials of (at most) second degree [e.g. \textit{T. L. Saaty}, Modern nonlinear equations, McGraw-Hill, New York (1967), p.122]. The more general equation \[ f(x)-f(y)=(x-y)h(x+y) \] has been solved by \textit{J. Aczél} [in Tagungsbericht Funktionalgleichungen (2. Tagung), Math. Forschungsinst. Oberwolfach (1963), p. 14] and the (slightly) even more general equation \[ f(x)-g(y)=(x-y)h(x+y) \] by \textit{Sh. Haruki} [Am. Math. Mon. 86, 577-579 (1979; Zbl 0413.39003)] on \(\mathbb{R}\) and by \textit{J. Aczél} [Math. Mag. 58, 42-45 (1985; Zbl 0571.39005)] on arbitrary fields of characteristric different from 2. This change of domain in the latter paper is not mentioned in the present work but there is a sentence about that equation and a similar one having been ``investigated on groups'' by \textit{M. Sablik} [C. R. Math. Acad. Sci., Soc. R. Can. 14, 207-212 (1992; Zbl 0796.39014)]. Note: Eq. (*) with the arithmetic mean replaced by other mean values and dual equations have been dealt with by \textit{J. Aczél} and \textit{M. Kuczma} [Aequationes Math. 38, 216-235 (1989; Zbl 0688.39006)] and by \textit{M. Kuczma} [Aequationes Math. 41, 33-54 (1991; Zbl 0726.39003)]. The subject of the main body of the present paper, inspired by the Elementary Problem E3338 posed by \textit{W. Rudin} [Am. Math. Mon. 96 (1989), p. 641)], is the equation \[ f(x)-f(y)=h(sx+ty)(x-y) \] \((x,y\in G\); \(f: G\to F\), \(h:G\to \text{Hom}(G,F))\), where \(G\) is a divisible Abelian group, \(F\) a field of characteristic zero and \(s,t\) are integers. Finally, a 5-page ``simple, direct'' solution of the generalization \[ f(x)-g(y)=(x-y) [h(x+y)+k(x)+\ell(y)] \] (\(x,y\in \mathbb{R}\); \(f,g,h,k,\ell\) real valued functions) is offered in place of the ``lengthy'' 8-page solution (including intermittent results on particular equations) in the book by \textit{P. K. Sahoo} and \textit{T. Riedel} [Mean value theorems and functional equations (1998; Zbl 0980.39015), pp. 100-108].
scientific article; zbMATH DE number 1945913